Soluble groups with few orbits under automorphisms
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Soluble groups with few orbits under automorphisms Raimundo Bastos1
· Alex C. Dantas1 · Emerson de Melo1
Received: 9 August 2019 / Accepted: 10 March 2020 © Springer Nature B.V. 2020
Abstract Let G be a group. The orbits of the natural action of Aut(G) on G are called “automorphism orbits” of G, and the number of automorphism orbits of G is denoted by ω(G). We prove that if G is a soluble group of finite rank such that ω(G) < ∞, then G contains a torsionfree radicable nilpotent characteristic subgroup K such that G = K H , where H is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that ω(G) = 3. Keywords Extensions · Automorphisms · Soluble groups Mathematics Subject Classification (2010) 20E22 · 20E36
1 Introduction Let G be a group. The orbits of the natural action of Aut(G) on G are called “automorphism orbits” of G, and the number of automorphism orbits of G is denoted by ω(G). It is interesting to ask what we can say about “G” only knowing ω(G). It is obvious that ω(G) = 1 if and only if G = {1}, and it is well known that if G is a finite group then ω(G) = 2 if and only if G is elementary abelian. Laffey and MacHale [4] proved that if G is a finite non-soluble group with ω(G) 4, then G is isomorphic to PSL(2, F4 ). Later, Stroppel [9], has shown that the only finite non-abelian simple groups G with ω(G) ≤ 5 are the groups PSL(2, Fq ) with q ∈ {4, 7, 8, 9} (see also [3] for finite simple groups with ω(G) 17). In [2], the authors prove that if G is a finite non-soluble group with ω(G) ≤ 6, then G is isomorphic to one of PSL(2, Fq ) with q ∈ {4, 7, 8, 9}, PSL(3, F4 ) or ASL(2, F4 ) (answering a question of Stroppel, cf. [9, Problem 2.5]).
The authors are supported by DPI/UnB and FAPDF-Brazil.
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Raimundo Bastos [email protected] Alex C. Dantas [email protected] Emerson de Melo [email protected]
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Departamento de Matemática, Universidade de Brasília, Brasilia, DF 70910-900, Brazil
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Geometriae Dedicata
Some aspects of automorphism orbits are also investigated for infinite groups. Schwachhöfer and Stroppel [8, Lemma 1.1], have shown that if G is an abelian group with finitely many automorphism orbits, then G = Tor(G) ⊕ D, where D is a torsion-free divisible characteristic subgroup of G and Tor(G) is the set of all torsion elements in G. In [1, Theorem A], the authors proved that if G is a FC-group with finitely many automorphism orbits, then the derived subgroup G is finite and G admits a decomposition G = Tor(G) × A, where A is a divisible characteristic subgroup of Z(G). For more details concerning automorphism orbits of groups see [9]. If G is a group and r is a positive integer, then G is said to have finite rank r if each finitely generated subgroup of G can be generated by r or fewer elements and if r is the least such integer. The next result can be viewed as a generalization of the above mentioned results from [1] and [8]. Theorem A Let G be a soluble group of finite rank. If ω(G) < ∞, then G has a torsion-free radicable nilpo
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